advances in numerical coding of the two fluid heii model 09/10/2013chats as 2013 rob van weelderen1...

CHATS AS 2013 Rob van Weelderen 1

Advances in Numerical Coding of the Two Fluid HeII Model

09102013

Rob van Weelderen CERNCyprien Soulaine IMFTMichel Quintard IMFT-CNRSBertrand Baudouy CEA-SaclayHerveacute Allain non-affiliated


Introduction (14)bull The simulation of superfluid helium (HeII) behaviour

is not always obvious and gives rise to specific numerical problems These problems often occur at the boundaries in regions with heat sources and due to thermal-acoustic waves

bull In a joint effort between the Institut de Meacutecanique des Fluides de Toulouse (IMFT) CERN and CEA-Saclay we have embarked on a project to develop a HeII simulator The aim of this project is to develop a code able to simulate the superfluid flow motion and its thermal interactions with superconducting magnet coils

09102013


Introduction (24)bull In a near future this tool will provide a solid basis

to develop the theory of superfluidity in porous media with the hope that it could significantly enhance the design of new devices cooled by He II

bull We report on the mathematical approach taken to numerically solve the 2-fluid model of HeII in laminar and turbulent regimes the specific problems encountered and the solutions found so-far We compare the results with relatively simple configurations for which analytical solutions exist and with data from an experiment on forced flow of HeII

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Introduction (34)bull Starting point is the two-fluid model by Landau

Khalatnikov which form a set of strongly coupled equations

bull Only few multi-dimensional codes reported in literature most report numerical difficulties which led them to either go 1-D or to simplify by dropping terms in the momentum equation (Kitamura et al Roa et al Zang et Al Ramadan and Witt)

bull Others assuming mutal friction and thermomechanical terms are approximately equal derive simplified equations (Suekane et Al Pietrowicz and Baudouy)

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Introduction (44)bull Tatsumoto et al proposes a numerical segregated

solution Bottura and Rosso a coupled algorithm

We propose an alternative segregated approach and extension of the ldquoPressure Implicit Operator Splitting (PISO)rdquo algorithm by Issa (1985) to ldquoSuper-PISOrdquo

The Pressure equation is directly derived from total mass balance and both momentum equations Its at the core of HeIIFOAM the code developed using OpenFOAMreg09102013


Mathematical model and assumptions 14

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Mass conservation Introducing the rate Γ at which superfluid transforms into normal assuming the rate of normal to superfluid to be symmetric (rsn = -rns)

Γ is an unknown variable to be solved



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Momentum balance equations

Represents the thermo-mechanical force which occurs when a temperature gradient exists

Gorter-Mellink mutual friction termwhich is present in the equations only at high velocities

momentum transfer between the two fluids Landau and Lifchitz (1969) suggest that there is no momentum transfer while Woods (1975) who derived superfluid hydrodynamic equations from a thermodynamic point of view keeps it

Mathematical model and assumptions 34Energy balance equation

To obtain a T equation we assume s and ρ independent of pressure

Mathematical model and assumptions 44Boundary conditions for surfaces

normal component of the total mass flux must vanish at any surface at rest

tangential component of the normal velocity v1048576 must be zero at a solid surface

In the absence of heat transfer between the solid surface and the fluid neglecting kn we get a no-slip boundary condition for the normal velocity

at a heated surface

At the helium bath entrance temperature and pressure are fixed values


Solutions and problems encountered (1

Segregated approach and extension of the Pressure Implicit Operator Splitting (PISO)rdquo algorithm by Issa (1985) to ldquoSuper-PISOrdquo The Pressure equation is directly derived from total mass balance and both momentum equations Its at the core of HeIIFOAM the code developed using OpenFOAMreg

Details in ldquoA PISO-like algorithm to simulate superfluid helium flow with the two-fluid modelrdquo Computer Physics Communications (Elsevier)

09102013



Heated wall gave many convergence issues and we circumvented those by considering the heat source as a volume heat source in the energy equation that is distributed in some cells

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Where α is a phase indicator equal to 1 for the heated cells and 0 elsewhere

This trick allows the heat flux to be relaxed in several cells which improves the stability of the calculation In particular the creation of normal fluid component is now spread over several cells instead of the singularity at the heated surface



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Capillary containing helium II in Landaursquos regime

This test consists in the simulation of a capillary (2h x L = 1 mm x 15 mm) tube filled with He II heated at the left-hand side and connected to a He II bath at the right-hand side

Initially superfluid and normal components are at rest at bath temperature and pressure field is uniformly zero The left-hand side is heated to q = 1000 Wm2



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Capillary containing helium II in Landaursquos regime steady state

To avoid or limit the propagation of the heat waves the left-hand side is gradually heated with a relaxation time of 10-2 s

At steady state this test-case has analytical solutions (Landau and Lifshitz (1969)) and the simulation agrees very well with it



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Capillary containing helium II in Landaursquos regime transient

Heat wave propagation speed in agreement with Landau amp Lifshitz

However we have a strong almost un-damped reflection from the open-bath end need to develop alternative boundary conditions



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Capillary containing helium II in Gorter-Mellinkrsquos regime

No problems encountered in this case theory and model agree perfectly (details skipped)

Temperature profile at steady state



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Fuzier and van Sciverrsquos experiments

Transient heat transfer for a forced flow of He II at high Reynolds number following the setup of the experiments performed by Fuzier and Van Sciver (2008)

They measured temperature profiles in a forced flow of superfluid helium in a 1 m long 98 mm inside diameter smooth tube



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liquid is pushed from a bellows pump to reach a velocity up to 22 ms

A heater is placed between 030 m and 031 m from the left-hand side

Liquid helium enters into the domain from the left-hand side at 2 ms and flows out at the right hand side of the tube Initially we assume that the superfluid and normal velocities are equal to 2 ms within the tube and the temperature is 17 K

Several probes are placed along the tube They correspond to the temperature measurements T3 to T8 of Fuzier and Van Sciver (2008) Simulations are performed with the full two-fluid model involving Gorter-Mellink mutual friction terms



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we start the heater at q = 99 Wcm2 for 20 ms and we record the temperature evolution until 250 ms for the different probes The results are in very good qualitative agreement with the experimental results shown in Fig 5 of Fuzier and Van Sciver (2008) the temperature peak for each probes are comparable both in amplitude and in time

2 ms



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8 ms



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16 ms


Superfluid flow through an area of beads

bull A capillary that contains 16 solid beads (05 mm diameter)

bull The heater is placed in a small extension of the capillary at the left-hand side of the array of beads

bull The capillary is initially filled with He II at 2 K The left-hand side is then warmed up to 104 Wm2 and the flow is simulated using the Gorter-Mellink model

09102013



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temperature profile at steady state in a capillary tube filled by 16 beads

The presence of solid materials inside the capillary leads to an increase of the ∆T (30 x more than without)



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normal velocity magnitude and of the normal velocity vectors in two adjacent REV We clearly notice cyclic flow patterns

This suggests that one may define a Representative Elementary Volume (REV) of the pore-scale physics and then apply volume-averaging methodology in order to derive macro-scale equations



09102013

superfluid velocity magnitude and of the superfluid velocity vectors in two adjacent REV We clearly notice cyclic flow patterns



Conclusion

1 Wersquove shown the first results mostly of validations of the use of a HeIIFOAM solver

2 Some problems still need to be accounted for especially on the open bath boundary

3 Mostly however good agreement can be found

4 Aim to apply this to derive macro scale equations for superconducting porous cables

5 Cross-check with dedicated tests to be foreseen

09102013

Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)






Solutions and problems encountered (1 (2)











Superfluid flow through an area of beads (2)



Conclusion


Introduction (14)bull The simulation of superfluid helium (HeII) behaviour

is not always obvious and gives rise to specific numerical problems These problems often occur at the boundaries in regions with heat sources and due to thermal-acoustic waves

bull In a joint effort between the Institut de Meacutecanique des Fluides de Toulouse (IMFT) CERN and CEA-Saclay we have embarked on a project to develop a HeII simulator The aim of this project is to develop a code able to simulate the superfluid flow motion and its thermal interactions with superconducting magnet coils

09102013





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at a heated surface






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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion





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at a heated surface






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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion






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at a heated surface






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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion








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at a heated surface






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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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at a heated surface






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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion







at a heated surface






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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion





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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion




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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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2 ms



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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8 ms



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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16 ms






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion






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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion



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Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion


Conclusion






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Slide 1

Introduction (14)

Introduction (24)

Introduction (34)

Introduction (44)




















Conclusion

advances in numerical coding of the two fluid heii model 09/10/2013chats as 2013 rob van weelderen1...

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